Algebra worksheet — GCSE Higher

1.A cyclist accelerates uniformly from rest to 6 m/s in 4 seconds, travels at a constant 6 m/s for 10 seconds, then decelerates uniformly to rest in 3 seconds. Work out the total distance the cyclist travels.

2.Which of these is an identity?

3.A phone company works out a monthly bill, £B, using the formula B = 18 + 0.05t, where £18 is the fixed monthly charge and t is the number of extra text messages sent beyond the free allowance, each charged at 5p. Farida's bill for one month is £24.50. Work out the number of extra text messages, t, she sent.

4.A company's weekly profit, P thousand pounds, when it makes x thousand items is modelled by P = x² − 8x + 12, for x ≥ 0. Work out the values of x for which the company makes a loss.

5.A market trader builds a display of oranges in a rectangular block. The block is n oranges wide and 4 oranges longer than it is wide. The number of oranges needed is 5 when n = 1, 12 when n = 2, 21 when n = 3, 32 when n = 4, and 45 when n = 5. Work out an expression, in terms of n, for the number of oranges needed for a display of width n.

6.Solve 6x − 5 = 3x + 13.

7.An allotment is in the shape of a rectangle. Its length is 5 m more than its width, x metres, and its area is 20 m². This gives x² + 5x − 20 = 0, which can be solved using the iterative formula xₙ₊₁ = 20 ÷ (xₙ + 5). Taking x₀ = 2, so that x₁ is the value found after the formula has been used once, work out x₂ correct to 2 decimal places.

8.p = 4. Work out the value of 2p³.

9.The tangent to a curve at the point (6, 1) has gradient −3. Work out the y-coordinate of the point where this tangent crosses the y-axis.

10.Work out the coordinates of the point where the graph of y = 2x − 6 crosses the x-axis.

11.A circle has centre (0, 0) and passes through the point (5, 12). Work out the equation of the circle.

12.A student is asked to simplify (6x³ − 15x²)/(4x² − 10x) fully. Four attempts are shown below. Which one is correct?

13.A theatre's front row has 18 seats. Each row behind has 4 more seats than the row in front. Which row has exactly 62 seats?

14.Write down the coefficient of xy in the expression 4x²y − 7xy + 2.

15.A sequence has nth term 3n + 4. Two terms are missing from the list of its first five terms: 7, 10, _, 16, _. Work out the missing 3rd and 5th terms added together.